We note that \(y_i^t\in \mathbb {R}^d\)  is the local dual variable held by client \(i\)  and \(\rho >0\)  is the coefficient of the quadratic term. Both the dual variables and the quadratic term serve to strike a balance between updating model parameters using local data while staying consistent with the server model (which serves to incorporate information from all participants). We note that FedProx [1]} similarly solves (REF ) with \(y_i^t\equiv 0\) . While this provides some level of safeguard against client drift, competitive performance of FedProx relies on careful tuning of \(\rho \)  [1]}, [3]}. The addition of the dual variables helps FedADMM to achieve automatic adaptation to heterogeneous data distributions and significantly alleviates the problem of hyperparameter tuning; we postpone detailed discussions on dual variables to the end of this section. In addition to the local model update (Alg. REF , lines 14-19), the dual variable for each selected client is updated in line 20. We combine the primal and dual variables to what we call augmented model \(\big (w_i + \frac{1}{\rho } y_i\big )\) , and use \(\Delta _i^t\in \mathbb {R}^d\)  to denote the update message of client \(i\)  to the server, that is the difference between successive augmented models:
\(\Delta _i^t = \left(w_i^{t+1}+\tfrac{1}{\rho }y_i^{t+1}\right)-\left(w_i^t+\tfrac{1}{\rho }y_i^t\right). \) 
